On Ramanujan's quartic theory of elliptic functions (Q5933399)

Let \(\varphi(q):=\sum_{n=-\infty}^\infty q^{n^2}\). In the classical theory of theta-functions, a fundamental inversion formula is NEWLINE\[NEWLINE{}_2F_1(\tfrac 12, \tfrac 12;1;x)=\varphi^2(q_2),\tag{1}NEWLINE\]NEWLINE where the relationship between \(q_2\) and \(x\) is given by NEWLINE\[NEWLINEq_r=q_r(x)=\exp\left(-\pi \csc(\pi/r)\frac {{}_2F_1(\frac {1}{r}, \frac{r-1}r;1;1-x)} {{}_2F_1(\frac {1}{r}, \frac{r-1}r;1;x)}\right)NEWLINE\]NEWLINE with \(r=2\). Ramanujan suggested that there should be corresponding theories when \(r=3\), \(4\), or \(6\); in this paper the authors develop the theory in the case \(r=4\). In the second section, the authors develop quartic inversion formulas from which they obtain an analogue of (1) for \(r=4\). In particular, with \(\psi(q):=\sum_{n=0}^\infty q^{n(n+1)/2}\) and NEWLINE\[NEWLINEA(q):=\varphi^4(q)+16q\psi^4(q^2),NEWLINE\]NEWLINE they show that NEWLINE\[NEWLINE{}_2F_1(\tfrac 14, \tfrac 34;1;x)=\sqrt{A(q_4)}.NEWLINE\]NEWLINE In the third section they obtain representations of certain quartic theta functions in terms of Dedekind's eta function. In the last section they develop methods for deriving series for \(\frac 1{\pi}\) associated with the quartic theory.